 Solution

For a point in the first quadrant, $xy$ is the area of the axis-aligned rectangle with opposite vertices at the origin and $(x,y)$. Thus the maximum says that the largest such quarter-rectangle with its outer corner on the ellipse has area one.

Equivalently, reflect this rectangle in both axes. Its four vertices $(\pm |x|,\pm |y|)$ lie on the ellipse, and its full area is $4|xy|$. ThereforeIts width is $1/\sqrt{2}$ and its height is $4\sqrt{2}$.